Let’s accept this definition, though it’s important to note that no-one defending such a thesis would interpret “science” in a narrow sense, but would regard it broadly as including the gathering of empirical evidence and rational analysis and conceptualising about that evidence. Thus, “scientism” would not, for example, deny that historians can generate knowledge, it would instead claim that they are doing so using methods that are pretty much the same as those used also by scientists. The differences in approach then arise from the pragmatics of what sort of evidence is accessible, not from their being distinct and separate “ways of knowing”.

The philosophical case that Hall presents is based on the problem of induction. No amount of observing a regularity proves that it will still hold tomorrow. The supposition that it will requires a “uniformity of nature” thesis that the future will be like the past, and since we cannot obtain empirical evidence from the future, that thesis — it is claimed — cannot be proven by science.

Hall then argues that science finds this “Past–Future Thesis” indispensable, but declares:

… either the PFT can be justified on non-empirical grounds, or it cannot be justified at all. If we accept the first horn, then we are conceding that scientific observation is not the only source of knowledge, and thus that scientism is false.

Hall then declares that the PFT is indeed true, and says:

… since there is no empirical way of defending PFT, we are forced to conclude that defending the assumption — and ultimately defending science itself — must rest on a philosophical foundation rather than an empirical one. And, thus, it follows that the claim that science is the only source of knowledge is false.

He then, rather derisively, declares this to be basic stuff akin to “remedial pre-algebra”, and finishes with: “If popular science writers wish to defend scientism, they would do well to demonstrate a modicum of understanding of the best arguments against their position”.

So, according to Hall’s argument, science is not the only source of knowledge because: (1) we know that the PFT is true, and (2) we know that from philosophy rather than from science.

But strikingly absent from Hall’s article is any philosophical defence of PFT. If one wants to use this example to show that philosophy can produce knowledge where science cannot, one first has to show that philosophy proves the PFT true. Yet Hall does not do this.

So this refutation of scientism fails right there. Showing that science cannot answer a question is only halfway to a refutation of scientism, since one then needs to show that some “other way of knowing” can produce a reliable answer.

But can the use of induction be defended? Personally I think it can, though as a matter of probability and likelihood, not of rigorous proof. (But then it is accepted that science never produces absolute proof, but only provisional, most-likely models that are better than any known alternatives.)

Hall indeed considers this, suggesting that: “… if we look at the past, we see that the future resembles the past all the time, so there’s an overwhelming probabilistic case for the PFT”, but then objecting that: “in appealing to what’s happened in the past as a guide to what will happen in the future, the would-be defender is assuming the very thing in question”.

But, we can consider the set of all events, past and future. And we can consider picking from that set, and encountering a sequence of picking one thousand white balls in a row and then the next ball being black. Obviously, the likelihood of that happening will depend on the probability distribution governing picking from the set, and — ex hypothesi — we don’t know that, since we don’t know about future events. But, that sequence will have some probability, and so we can consider the ensemble of all possible probability distributions.

If there are long periods of stasis of unknown length, it is more probable that one is somewhere within the period of stasis rather than exactly at its end. That follows simply because there is only one “slot” at the end of the sequence but lots of slots that are not at the end. Given a long sequence of normality, and picking our location on that sequence at random, it is more likely that we will be somewhere boring in the midst of the sequence, rather than at the highly particular “last day of normality” right at its end. In essence, we’re not using the past as a guide to the future, we’re using it as a guide to the present time, and asking whether it is unusual.

This analysis requires as to conceptualise a birds-eye overview of the timeline, but it doesn’t require any assumption about the future and it doesn’t require knowing the probability distribution of future events.

Of course it is no guarantee, and for all we know the probabilities could be such that normality is coming to an imminent end. But, the sub-set of probability distributions that make it likely that, after having picked a thousand white balls in a row, the next is a black, is much smaller than the set of all possible probability distributions. Only a very special and particular probability distribution could make it more likely that we are exactly at the end of such a sequence, rather than anywhere else along it. And, given that we don’t know the probability distribution, that is unlikely. So it is more likely than not that a sequence of stasis will continue with the next pick.

Again, this argument does not depend on assuming a uniform probability distribution, it only depends on their being a probability distribution, and on considering the super-set of all possible such probability distributions.

This line of reasoning has been proposed by Ray Solomonoff, who formalised and developed it into his “Formal Theory of Inductive Inference”. I’m not aware of any refutation of the argument and so I currently regard it as a sufficient resolution of the problem of induction. (Though part of the point of writing a blog piece about it is that, if it’s wrong, someone might tell me why!)

As regards scientism, a last question arises as to whether the above argument counts as “science” or as “philosophy”. It is certainly a rational analysis involving mathematical reasoning. It is not a rebuttal that can be observed empirically with a pair of binoculars or a microscope. But then no sensible account limits science to what can be directly observed. That’s only the half of it. Science is just as much about the concepts and rational analysis that make sense of the empirical world. Thus the above rebuttal is squarely within the domain of science, and so the attempt to defeat scientism fails.

40 thoughts on “Does the problem of induction defeat scientism?”

“Again, this argument does not depend on assuming a uniform probability distribution, it only defends on their being a probability distribution, and on considering the super-set of all possible such probability distributions.”

I would like to agree, but cannot. You have assumed that the probability distribution, or probability distribution of probability distributions, or… to infinite regress applies as well to what we have not yet observed, as to what we have. But that is precisely what needs to be established.

But I haven’t made any assumptions about what those probability distributions are, I’ve only assumed that one can meaningfully talk about those probability distributions, which is really only the assumption that one can talk meaningfully about the future.

As usual, the critic of science confuses a working assumption with a metaphysical presumption. Science doesn’t “know” the past is a guide to the future, it just points out that either it is, or there is no guide! Science posits that there may be laws which hold across time, and attempts to discern good candidates for those laws. Maybe there are no such laws! But then you couldn’t have done any better than the scientific approach since outcomes would be fundamentally random anyway.

Thanks for the nice description of a too common argument against scientism.

I am continually puzzled by philosophers and even philosophers of science who appear to think science is built on “premises” such as “past predicts the future”. Science is a web of ideas that best explains what is observed, nothing more about premises required. What many philosophers consider science’s “premises”, such as PDF, I see as accepted hypotheses based on observation. Most scientists would be excited and delighted to find examples of the past not predicting the future, such as a changing speed of light, law of gravitation, or particle mass.

There also is the troublesome further vague claims of “other ways of knowing”.

Perhaps we could say all other ways of knowing are subjective? (I am including logic and mathematics as part of objective science.)

Agreed. A widespread claim is that science rests on philosophical assumptions. But, actually, science is pragmatic, and the “scientific method” has evolved out of trying things out and finding out what works. The ultimate justification is thus not “philosophical underpinnings” but “it works”, as demonstrated by the fact that technology built on science does work.

Mark Sloan: “Science is a web of ideas that best explains what is observed, nothing more about premises required.”; as soon as you attempt to say what you mean by “best”, you are doing philosophy. (And as soon as Richard Wien invokes “the best”, he is offering a justification.)

“mathematics as part of science”; Coel would agree, but I find that a stretch. It comes close to making scientism trivially true by extending the boundaries of what we call science to include all objective knowledge. There is no physical fact corresponding to the mathematical fact that the square root of 2 is irrational. The arithmetic of counting bricks is, as Coel will point out, an observational demonstration of a physical system to which the postulates of arithmetic apply, but there are areas of mathematics (such as the mathematics of the transfinite) that have no such physical application.

It comes close to making scientism trivially true by extending the boundaries of what we call science to include all objective knowledge.

The substance is then the claim that all knowledge forms a unified whole, and is not divided into discrete and fundamentally different “ways of knowing”.

As you suggested, I will indeed claim that mathematical statements such as “2+2=4” are adopted as real-world models (akin to “laws of physics”). If the claim is that they rest instead on axioms, not empirical reality, I then claim that the axioms themselves are chosen as real-world models.

There is no physical fact corresponding to the mathematical fact that the square root of 2 is irrational.

But one can equally use physics to make statements about things that aren’t directly observable. The structure here is: empirical reality => physical models of the world => implications of those models.

One can equally suggest: empirical reality => *mathematical* models of the world => implications of those models.

“But one can equally use physics to make statements about things that aren’t directly observable.” When it comes to theories that canot give observable consequences, such as (so I read) string theory, then people do indeed discuss if we are still within the domain of science.

As for the inverse square law, we can observe its consequences, note when they are an imperfect fit to observation, as in the orbit of Mercury, and infer that the law needs modifying. Whereas there are no observable consequences, or possible modifications, to the irrationality of sqrt 2.

Take the example of a list of solar eclipses that occurred (as view-able from somewhere on Earth), between one million years ago and two million years ago. There is no way of verifying any of these events by observation. But it seems to me perverse to then declare that such predictions are “not science”, since they are implications of a real-world model where the correspondence of the model to the real world can be tested sufficiently well to then have confidence in the model and thus in the list of eclipse predictions.

The status of mathematical theorems is then similar. They are implications of a real-world model where the overall model is verified as being a good real-world model. Some of the implications would then be directly observable but some would not.

Very interesting. We would agree that statements about eclipses two million years ago are true or false, and we would get it right or wrong depending on how competently we did the calculations. What about eclipses two billion years ago, which may well not be amenable to computation evenin principle because of the chaotic gravitational effects of asteriods, or something? I would maintain that the statements would be true or false but we would never be able to tell which. I think you’d agree. So it is not necessary for a consequence of assumptions to be calculable, for that consequence to be true. (That doesn’t refute you, since you merely said it was sufficient, but I do find it suggestive).

Now try this: say some playful mathematician dreams up a non-trivial set of axioms for which there is no physical model (this may actually have happened, for all I know). I would say that we could still identify theorems within that system, and talk about true, false, or perhaps indeterminate statements within it, even though we had no recourse at any stage to observation. This makes me, when it comes to maths, a reluctant Platonist. Rescue me if you can

I would maintain that the statements would be true or false but we would never be able to tell which. I think you’d agree.

Yes, agreed. It’s easy to think of questions about which there is a truth of the matter, yet we can never know it because the universe does not contain the information. An example is the question “What was the name of the person who fired the arrow that hit Harold in the eye at Hastings?”. Presumably there is a fact of the matter (even if it is “there was no such event”), yet we can never know it.

… say some playful mathematician dreams up a non-trivial set of axioms for which there is no physical model … we could still identify theorems within that system, and talk about true, false, or perhaps indeterminate statements within it …

My reply would be two-fold:

(1) I would deny that there could be any such thing as “axioms for which there is no physical model” because they could not have any meaning.

We can of course envisage “science fiction”, by which we mean variations on observable reality. So we could envisage an Earth that has twice the mass, where the dominant life-forms were lizards, which breathed a methane atmosphere.

But, all of those concepts (“twice”, “mass”, “dominant”, “life-form”, “lizards”. “breathed”, “methane” and “atmosphere”) are concepts that only have any meaning to us owing to their reference to features of the external world.

I also claim that the same is true about basic maths or logical axioms. “1”, “plus”, “two”, “equals” and “four” are also concepts that have meaning only owing to real-world reference.

So we can contemplate a mathematical system in which “2+2=5”, but the only way that even has any meaning is if *most* of the concepts in that system have real-world meaning and that means real-world correspondence.

As with science fiction, we can vary it a bit, but it is still largely about our empirical world. We cannot make it entirely unrelated to the observable world.

If we tried, we’d write down a series of sqiggles on some paper, but quite literlly they would have no meaning except through real-world reference. So, you’d have something like:

6fk””f o3%3k t2d-d-d-d* fjwjwjs 2d70b**8=

How would you “identify theorems” in that, and discuss whether they were “true”, if you didn’t know what any of it meant?

So, I would assert that any such system (that could be meaningfully interpreted) would still be about our empirical world, even if it were a science-fiction variation on that world, with variations that are still about real-world concepts.

(2) Let’s ignore all that and suppose we did indeed have and understand some mathematical system that is entirely independent of the world. In order to talk about whether theorems in that system are true within that system, we’d need some way of establishing truth within that system. How would we do that? Well, we could try using *our* logic, based on our real world. But, that is, of course, invalid.

So, we’d have to use logic from within that system. But how could we establish that the logical reasoning we used was valid? I don’t think we could do that. (And Godel tells us that we couldn’t prove the self-consistency of that system from within that system; though of course we’re not allowed to assume that Godel holds in that system.)

Well, let’s ask how we know that *our* logic is valid. That answer is that we know it is valid because it works applied to the real world. That, ultimately, is the justification for our mathematical and logical systems. (Again, Godel tells us we can’t justify them internally.) It’s the same validation that tells us that science produces a pretty good model of the external world — our technology works.

Thus, our logic/maths is arrived at as a real-world model, and is justified and validated by being a sufficiently good real-world model. That, indeed, is the only basis on which you can be confident that the reasoning that tells you that sqrt(2) is irrational is valid. That’s why it is ultimately an empirical truth, about the logical/mathematical structure of our physical world, rather than a purely conceptual truth about Platonic entities.

If scientism Is only epistemology and not ontology, that resolves issues such as mitochondrial Eve, ancient eclipses, or my sugar cube. But we seem no nearer than ever to agreement about mathematics.

“How would you “identify theorems” in that, and discuss whether they were “true”, if you didn’t know what any of it meant?”

Given the multiplication table for the abstract group D3d, one could verify that a particular set of matrices obeyed that multiplication, without realising that these matrices had any physical meaning. (I was very much in this situation when for some strange reason I had to learn about determinants for A-levels – yes, I’m that old.)

It is of course true that all mathematical reasoning involves logic, and that true statements about things in the world are governed by logic. but I continue to maintain that the primary meaning of a mathematical system resides within the axioms and transformation rules, which may or may not be explicit. Some real-world objects are then modelled by this or that * interpretation* of the formal system; others are not. There are several kinds of geometrical system that are modelled by the point group Oh (both regular octahedra and regular cubes, for example), and many kinds of physical obect that (ideally) show such geometries, but that does not provide a test for the mathematical system. The theorems regarding Oh are not verified by the fact that [SiF6]2- is described by this group, nor are they falsified by the fact that SiF4 (highest symmetry C2v) is not.

But if we’re turning to ontology, what does a Platonist even mean by asserting that concepts “exist”? As far as I can work out, the only actual attribute being asserted is that such things are self-consistent.

But I can self-consistently conceive of unicorns, yet they are considered not to exist. If the Platonist replies that yes they exist, as concepts, then I think that’s just empty word play. One could equally say that the make-believe world in which my nephew has just scored a hat-trick in the World Cup final also “exists” conceptually.

I continue to maintain that the primary meaning of a mathematical system resides within the axioms and transformation rules, …

Let’s look at some axioms, such as some of Peano’s axioms:

1. 0 is a natural number.
2. For every natural number x, x = x.
3. For all natural numbers x and y, if x = y, then y = x.

In order for these to even have meaning, we need to already know a lot of concepts, including what is mean by: “is a”, “for all”, if … then”, “and” and “equals”. We also need agreement that the symbols “x” and “y” can stand for any number, et cetera.

Thus, the meaning is not contained in the axioms. Their meaning rests in a whole web of understandings which, I assert, comes from us humans having adopted a basic understanding of things as a real-world model.

“In order for these to even have meaning,..” This takes us to the core of our disagreement. The natural numbers are one set that obey Peano’s axioms. There may be others. Specifically, there are sets of matrices that represent (i.e. obey the rules that define) Oh, these matrices can be used to represent a set of rotations and reflections in three-dimensional Euclidean space, and those rotations and reflections themselves can be used as an interprettion of that group. No doubt there are many other representations.

You maintain that all this mathematics is true by virtue of the fact that there exist physical interpretations. But given the fact that there is commonly more than one interpretation, and that it is conceivable that we could construct mathematically consistent objects that have no interpretation whatsoever, this seems to me artificial.

Regarding existence, this puzzled me for a while, until I remembered Russell’s dictum that existence is not a predicate. To say that X exists is merely to say that there are meaningful statements that can be made about X, not that X has some additional property of “existing”. If I remember this, I am left with a very weak form of mathematical Platonism, safely insulated from observation unless, as a matter of physics, there are objects that represent the mathematical system.

You may suspect, as I do, that the difference between us is now just a matter of semantics. In any case, I thank you for forcing me to think it through.

You maintain that all this mathematics is true by virtue of the fact that there exist physical interpretations.

Yes, or more basically, that it only has meaning owing to an interpretation in terms of real-world concepts. (And yes, the truth of that real-world meaning can then only be verified using real-world concepts.)

But given the fact that there is commonly more than one interpretation, …

I suggest that that *supports* my claim, since it implies that the meaning is in the interpretation, and is not intrinsic to the squiggles and symbols. Also, normal-language words can often have multiple interpretations, and again the meaning is in the interpretation.

… and that it is conceivable that we could construct mathematically consistent objects that have no interpretation whatsoever, this seems to me artificial.

That I would deny. There would be no “maths”, no meaning to it. How would you distinguish such a mathematical object from the result of a thousand scrabble pieces thrown into the air and let fall? How would you verify that the mathematical object is self-consistent?

To say that X exists is merely to say that there are meaningful statements that can be made about X, not that X has some additional property of “existing”.

Hmm, so Narnia, Middle Earth, and Discworld all exist? The make-believe world in which my nephew scores a hat-trick in a World Cup final does actually exist? This makes the concept of existence empty. OK, so maths does exist in that sense, so do ghosts, unicorns, God and faeries.

My concept of “existence” clearly needs to be totally rethought. One school of analytical philosophers got round the problem you point out by saying that if X does not exist, then all statements about X are meaningless because of lack of referent. But that merely leads us on to an issue where I continue to stand my ground, namely meaningfullness.

If I write down the multiplication table of an abstract group, I would say that the meaning of what I have written is just that, the group’s multiplication table. It does not matter whether or not there are any physical systems whose transformations obey that multiplication table; I can use the table to combine operations, if I do so accurately then what I say is true, and if I make a mistake then it is false. You say that the group only acquires meaning when it is shown that there is some physical system whose transformations obey that multiplication table. I don’t see how we can resolve this.

Imagine that I have found a group which correctly describes the transformations of some set of objects in 17-dimensional Euclidean space, but nothing else. You could fairly say either that it isn’t meaningful (because there is no such thing as 17-dimensional Euclidean space) or that it is (because we can use methods validated in 3-dimensional space to predict logically what that space would have to be like). I would say that it is meaningful anyway.

I don’t see how we can resolve this. Unless either of us has a novel insight, I suggest we leave it there.

If I write down the multiplication table of an abstract group, I would say that the meaning of what I have written is just that, the group’s multiplication table. […] You say that the group only acquires meaning when it is shown that there is some physical system whose transformations obey that multiplication table.

I would consider this in two parts: (1) the concept of multiplication, and (2) the thing being multiplied.

Even if you have a totally abstract thing-being-multiplied, that has no physical interpretation, then I would assert that the operation, the multiplication, does have meaning, because that multiplication is an operation derived from real-world models and it obtains its meaning from that.

So, yes, it is meaningful, but the meaning is in the operation of multiplication, not in the thing being multiplied.

To take an example, consider: 2a + 2a = 4a (which combines addition, equality and multiplication). This is meaningful, even if I attach no meaning to the symbol “a”. Indeed, the whole point of such maths is to abstract the meaning of the operation, separating it from any meaning of the symbol “a”. So it works whether “a” refers to a cow or to an abstract mathematical set.

So I don’t think your example refutes my basic claim.

By the way, being boring and scientistic, I’d just go with the naive idea that to “exist” is the same as to “physically exist”, in that only things with physical existence do actually “exist”.

That still allows the existence of complex patterns that supervene on physical stuff (a “cow” or a “police force” or a “society” are examples). Concepts also exist (as physical neural-net patterns in the brain of the person whose thinking instantiates the concepts), but the things being conceived (unicorns, or mathematical structures) needn’t themselves “exist” and don’t if they are not material.

If you can think of an obvious refutation of that line I’d be interested, but so far that’s how I see “existence”.

There must have been a “mitochondrial eve”, a female human from whom all modern humans descend, such that if we trace back our mother to grandmother to great grandmother, and keep going, everyone alive today gets back to that one individual.

How do we know this? Well, it is not from direct observation. It is not the case that we’ve traced the genealogy of every human alive, going back hundreds of thousands of years, and shown that every line does indeed trace back to one individual. That would be utterly impossible. The “mitochondrial eve” (ME) is not observable. Yet the existence of an ME is not doubted, and no-one queries that the concept is a scientific one.

Instead, we deduce the existence of a ME conceptually. Given what we know about biology, an ME is necessary (= logically entailed).

Thus the statement that there must have been an ME is similar to the status of the statement that sqrt(2) is irrational. Both are arrived at by logical deduction from an accepted theory, not by (direct) empirical means.

Re mitochondrial Eve, I think the impossibility is in practice but not in principle. Just like knowing the exact number of molecules of sugar in an isolated sugar cube is impossible in practice, even though there does exist one and only one correct answer. I don’t think it helps

But whether it is impossible to verify in principle, versus in practice, doesn’t seem all that important. “Scientism” is a claim about epistemology, and the more relevant point is how we know something is true. Thus mitochondrial Eve is akin to the irrationality of sqrt(2), in that both are deduced by reasoning within an accepted theoretical scheme, rather than being directly observed.

Obviously, how we define science can determine whether we conclude scientism is true. But dictionary definitions are just what words commonly refer to with multiple definitions that change over time being normal. It seems to me most useful to advocate for a definition of science that specifically includes logic and mathematics.

Such a definition may provide a clear, and useful, distinction between objective knowledge from science and subjective knowledge from other sources. Suggestions for other sources of objective knowledge than science + logic + mathematics) are welcome. Surely scientism’s critics are not thinking only of logic and mathematics as the “Other sources of knowledge”!

Of course, with this definition when philosophers are doing logic and mathematics they would them be doing science, but so what? From the Bertrand Russel quote I provided in another comment, I expect at least he would have no problem with that. And philosophers used to do all the science there was.

To me, it remains an open question whether our mathematics and logic are innate to our universe (meaning our physical reality). If they are innate to our physical reality, then it would be sensible to recognize that and include them as part of science.

But perhaps logic and mathematics are not innate in our universe. We still can include them in our definition of science if we think that is most useful for intellectual discourse.

Mark wrote:
‘ I am continually puzzled by philosophers and even philosophers of science who appear to think science is built on “premises” such as “past predicts the future”. ‘

Good point. The problem is that philosophers often have an overly abstract or metaphysical way of thinking about things. In fact I think Hall’s appeal to Hume is misguided, because Hume largely rejected that way of thinking and was perhaps the first philosopher to take a more naturalistic approach to philosophy.

‘The substance is then the claim that all knowledge forms a unified whole, and is not divided into discrete and fundamentally different “ways of knowing”.’

That’s a much better way of saying it than putting it in terms of “science” and then having to explain that you are using the word “science” in an unusual way. When you put it this way, I can enthusiastically agree with this statement (while still disagreeing with your philosophy of mathematics).

‘This makes me, when it comes to maths, a reluctant Platonist. Rescue me if you can.’

My main objection to mathematical platonists is that they usually engage in misguided talk about “mathematical objects”, sometimes even claiming that those objects reside in a “realm” of abstract objects. Perhaps you’re not that kind of mathematical platonist.

If the platonist’s claim is merely that mathematical statements can be objectively true, as opposed to being true only in some mind-dependent way, then I would agree.

I would say (contrary to Coel) that mathematical statements are best thought of as pure abstractions. But that doesn’t require saying that we reach them by a fundamentally different “way of knowing”. Or that they are statements “about abstract objects” (as platonists tend to say).

As for axioms, which were mentioned above… Sometimes mathematicians work from axioms. But axioms aren’t essential to mathematics. People engaged in counting and arithmetic for millennia before Peano came up with his axioms. Counting and arithmetic are the result of shared practices, not axioms. We copy from other people the practice of counting “one, two, three”, etc.

Still sounds to.me like you are trying to support logic with induction then support induction with logic.

Yes, in a sense. Science is not foundational; one cannot derive anything from first principles or a priori axioms. So it’s a web of ideas, which is iterated against empirical reality. So yes, the whole thing is bootstrapped, and each idea is then justified in terms of other ideas, with the ensemble justified by the demonstrable fit to reality. Given that, induction is no worse supported than any other part of the web. The puzzle only really arises if one attempts a building-from-foundations edifice for science, but that’s not going to work.

That I would deny. There would be no “maths”, no meaning to it. How would you distinguish such a mathematical object from the result of a thousand scrabble pieces thrown into the air and let fall? How would you verify that the mathematical object is self-consistent?

But I can’t see the real world correspondence of, for example, “therefore” …

The concept “therefore” is invoked by a brain that evolved to model the real world, a brain that developed by interacting with the real world through the senses, and it involves logical concepts that were created to make sense of the real world, and it is expressed in a language whose basis is about communicating about the real world.

For example it wouldn’t be impossible for a Boltzmann brain to wink into existence, think of some mathematics, understand it, draw some conclusions and then wink back out of existence. That brain wouldn’t even have been aware of the real world.

For example it wouldn’t be impossible for a Boltzmann brain to wink into existence, think of some mathematics, understand it, draw some conclusions and then wink back out of existence.

But for every brain doing that, there would be a near-infinite number of similar brains thinking gibberish and not “understanding” anything. The only way of distinguishing that one brain from all the others would be by referring to some anchor, such as the real world.

Even if that argument fails, the possibility of a fluke Boltzmann brain wouldn’t alter claims about mathematical and logical concepts invoked by human brains, and whether they arise as real-world models.